Like others have mentioned, pediatricians are specialists in children. Which has its advantages and disadvantages. They are definitely the ones to seek if your child has a chronic illness, if you're worried about something unusual or serious. They also are more likely to be the ones taking care of your children in the hospital although some family docs admit children as well.
Family docs have the advantage of continuity of care for the whole family, and because they often take care of mom too, they are much less likely to be adversarial toward the parents, like some peds can be. Their training is primarily well-children and less serious illnesses, but it also depends on the individual doctor and what training or interests he/she has pursued. Many family docs, especially those that deliver babies, will have a large number of children in their practice and will be quite comfortable with a broad range of pediatrics. Another advantage is that your children don't need find another doctor when they turn 18! It is true that family practice has much more of a "wellness" philosophy than other specialties, but again, a lot depends on how the individual doctor was trained.
It's hard to generalize about either specialty. Not all family docs are alternative or supportive of AP, trust me! And not all pediatricians are anti-AP. What's most important is finding a doctor that you feel comfortable with and trust!
And for the record, I'm a family doc who sees kids, and I really enjoy it. But our children's doctor is a pediatrician - we probably do disagree on some "parenting" things, but I don't go to her for parenting advice. I go to her because of her years of experience with really sick kids, and I know that if my kids have something rare or serious, she'll pick it up!
HTH!